# CDT Examples These examples reproduce key experiments from the Causal Dynamical Triangulations literature. Each script runs a Metropolis Monte Carlo simulation using the `caset` C++ library and produces plots that can be compared side-by-side with figures from the original papers. All scripts accept `--save ` to write output to disk, or display interactively by default. Run any script with `--help` for the full set of tunable parameters. ```{note} The paper results were obtained on lattices of $N_4 = 10\,000$--$362\,000$ four-simplices with $T = 80$ time slices and $10^5$--$10^8$ Monte Carlo sweeps. The plots shown here use $N_4 \sim 800$--$1\,600$ to keep runtimes under a few minutes. Observables that depend on large-scale geometry (phase boundaries, Hausdorff scaling, $\cos^3$ profile shape) require at least $N_4 > 5\,000$ to become visible. Where our small-lattice results already confirm the paper predictions, this is noted explicitly; where they do not yet, the expected large-lattice behaviour is described. ``` --- ## $N_{32}$ Distribution at Fixed $N_{41}$ **Script:** `examples/n32_distribution.py` \ **Reproduces:** Figure 2, Ambjorn, Jurkiewicz & Loll, *Reconstructing the Universe*, Phys. Rev. D **72** (2005) \[[hep-th/0505154](https://arxiv.org/abs/hep-th/0505154)\] **Paper result (Fig. 2):** At fixed $\tilde{N}_4 = N_4^{(4,1)}$, the distribution of $(3{,}2)$-simplices $P(N_4^{(3,2)})$ is sharply peaked, with the peak narrowing and shifting to larger $N_4^{(3,2)}$ as the target volume increases. This demonstrates that the volume-fixing term $\delta S = \epsilon |N_4^{(4,1)} - \tilde{N}_4|$ effectively constrains both simplex types simultaneously. **Our result:** **Confirmed.** The distributions are sharply peaked at each target volume ($\tilde{N}_4 \approx 193$, $383$, $686$) and shift to larger $N_4^{(3,2)}$ exactly as in the paper. The ratio $N_4^{(3,2)} / N_4^{(4,1)} \approx 2.3$--$2.5$ is consistent with the paper's Table 1. ```bash python examples/n32_distribution.py --n-meas 300 ``` ```{image} assets/cdt/n32_distribution.png :alt: Distribution of N_4^(3,2) at three target volumes :width: 70% :align: center ``` --- ## Spectral Dimension **Script:** `examples/spectral_dimension.py` \ **Reproduces:** Figures 9 and 10, Ambjorn, Jurkiewicz & Loll, *Reconstructing the Universe*, Phys. Rev. D **72** (2005) \[[hep-th/0505154](https://arxiv.org/abs/hep-th/0505154)\] **Paper result (Figs. 9--10):** The spectral dimension $D_S(\sigma) = -2\, d\log P(\sigma) / d\log\sigma$ rises monotonically from $D_S(0) = 1.80 \pm 0.25$ at short diffusion times to $D_S(\infty) = 4.02 \pm 0.1$ at large scales. The best three-parameter fit is (Eq. 29): $$ D_S(\sigma) = 4.02 - \frac{119}{54 + \sigma} $$ **Our result:** **Qualitatively confirmed.** The measured $D_S(\sigma)$ rises monotonically from near zero and approaches $\sim 3.5$ at $\sigma = 200$, matching the shape of the paper's curve. The absolute values are lower than the paper's because our lattice ($N_4 \approx 1\,000$) is much smaller than the paper's ($N_4 = 20$k--$80$k, $T = 80$); the spectral dimension is a finite-size-dependent quantity that converges upward with increasing volume. The paper's fit curve (red dashed) is overlaid for reference. ```bash python examples/spectral_dimension.py --n-simplices 800 --n-configs 8 --max-sigma 200 ``` ```{image} assets/cdt/spectral_dimension.png :alt: Spectral dimension D_S(sigma) with reference fit from the paper :width: 100% ``` --- ## Effective Action and Minisuperspace **Script:** `examples/effective_action.py` \ **Reproduces:** Figures 11, 12, 13, Ambjorn, Jurkiewicz & Loll, *Reconstructing the Universe*, Phys. Rev. D **72** (2005) \[[hep-th/0505154](https://arxiv.org/abs/hep-th/0505154)\] **Paper results:** - **Fig. 11:** The kinetic-term scaling dimension $D_2$ is estimated by minimising overlap error of rescaled volume-difference distributions. The minimum occurs at $D_2 \approx 2.12$, interpreted as $D_2 = 2$. - **Fig. 12:** The distribution of rescaled volume differences $z = |\Delta N_3| / N_3^{1/D_2}$ at $D_2 = 2$ collapses onto a Gaussian $e^{-cz^2}$ independent of $N_3$. - **Fig. 13:** The Monte Carlo volume-volume correlator matches the prediction from the minisuperspace effective action (Eq. 40). **Our results:** - **D_2 scaling (top left):** The overlap-error curve has its minimum near $D_2 \approx 2$, **consistent with the paper.** - **Volume differences (top right):** The measured distribution shows a decaying shape. The Gaussian fit captures the trend but scatter is large due to limited statistics at $N_4 \sim 800$. - **$\cos^3$ comparison (bottom left):** The Monte Carlo average profile is relatively flat compared to the peaked $\cos^3(\pi\tau/T)$ prediction. This is expected: the $\cos^3$ shape emerges only at $N_4 \gg 10\,000$ with $T \geq 40$ time slices. At our lattice size, the profile spans only $\sim 9$ slices with insufficient dynamic range to develop the characteristic bulge. - **Action evolution (bottom right):** The Regge action and four-volume track each other during equilibration, confirming the volume-fixing term drives $N_4$ toward the target. ```bash python examples/effective_action.py --n-simplices 800 --n-meas 60 ``` ```{image} assets/cdt/effective_action.png :alt: Effective action analysis: D_2 scaling, volume differences, minisuperspace comparison :width: 100% ``` --- ## Volume Profiles in Phases A, B, C **Script:** `examples/volume_profile_phases.py` \ **Reproduces:** Figures 4, 5, 6, Ambjorn, Jurkiewicz & Loll, *Reconstructing the Universe*, Phys. Rev. D **72** (2005) \[[hep-th/0505154](https://arxiv.org/abs/hep-th/0505154)\] **Paper results:** - **Fig. 4** (Phase A, $\kappa_0 = 5.0$, $\Delta = 0$): thin, elongated branched-polymer geometry with irregular maxima. - **Fig. 5** (Phase B, $\kappa_0 = 1.6$, $\Delta = 0$): crumpled geometry collapsed to $\sim 2$ time slices. - **Fig. 6** (Phase C, $\kappa_0 = 2.2$, $\Delta = 0.6$): extended de Sitter universe with $N_3(\tau) \propto \cos^3(\pi\tau/T)$. **Our result:** At $N_4 \sim 800$ the three profiles are **not yet differentiated** -- all show a roughly uniform distribution across $\sim 9$ time slices. The phase transitions in the paper occur at $N_4 \sim 20$k--$45$k; at smaller volumes the system is too small to develop the distinct geometric structures that characterise each phase. The dashed $\cos^3$ reference curve shows the expected shape for Phase $C_{dS}$ at large $N_4$. To reproduce the paper's Fig. 6 shape, run with `--n-simplices 10000` or larger (estimated runtime: several hours). ```bash python examples/volume_profile_phases.py --n-simplices 800 --n-therm 80 --n-meas 30 ``` ```{image} assets/cdt/volume_profiles_surface.png :alt: Surface-of-revolution plots of the universe in phases A, B, C :width: 100% ``` ```{image} assets/cdt/volume_profiles_profile.png :alt: Line plot of N_3(tau) for all three phases :width: 80% :align: center ``` --- ## Volume-Volume Correlator and Hausdorff Dimension **Script:** `examples/volume_scaling.py` \ **Reproduces:** Figures 7, 8, 12, Ambjorn, Jurkiewicz & Loll, *Reconstructing the Universe*, Phys. Rev. D **72** (2005) \[[hep-th/0505154](https://arxiv.org/abs/hep-th/0505154)\] **Paper results:** - **Fig. 7:** The rescaled volume-volume correlator $c_{\tilde{N}_4}(x)$ with $x = \delta / (\tilde{N}_4^{\text{eff}})^{1/4}$ collapses onto a universal peaked curve for system sizes $\tilde{N}_4 = 10$k, $20$k, $40$k, $80$k, $160$k. - **Fig. 8:** The overlap error has a clear minimum at $D_H \approx 3.8$, interpreted as $D_H = 4$. **Our result:** At $N_4 \sim 650$--$1\,900$ the correlator data points (top left) do not yet collapse onto a single curve, and the D_H estimate (top right) does **not** show a clear minimum at $D_H = 4$. This is expected: the paper explicitly notes that finite-volume effects are significant below $\tilde{N}_4 = 20$k and uses $T = 80$ time slices (vs. our $\sim 9$--$11$). The volume-difference distribution (bottom left) shows a decaying shape consistent with the Gaussian fit, in qualitative agreement with Fig. 12. To reproduce the scaling collapse, run with `--n-simplices 10000` or larger and `--n-meas 100+`. ```bash python examples/volume_scaling.py --n-simplices 800 --n-meas 40 ``` ```{image} assets/cdt/volume_scaling.png :alt: Volume scaling analysis with correlator and Hausdorff dimension :width: 100% ``` --- ## Phase Diagram **Script:** `examples/phase_diagram.py` \ **Reproduces:** Figure 3, Ambjorn, Jurkiewicz & Loll, *Reconstructing the Universe*, Phys. Rev. D **72** (2005) \[[hep-th/0505154](https://arxiv.org/abs/hep-th/0505154)\]; and the phase diagram from Gorlich, *Introduction to Causal Dynamical Triangulations* (2013) **Paper result (Fig. 3):** The $(\kappa_0, \Delta)$ plane splits into three phases -- A (large $\kappa_0$), B (small $\kappa_0$, small $\Delta$), C (moderate $\kappa_0$, $\Delta > 0$) -- separated by first-order transition lines. **Our result:** At $N_4 \sim 400$ per point the phase diagram is **uniformly Phase C** across the entire scanned region. The phase transitions are not visible because they require lattice sizes of $N_4 > 20\,000$ to resolve: at small volumes the system does not have enough degrees of freedom to collapse into Phase B or fragment into Phase A. The white star marks the de Sitter point $\kappa_0 = 2.2$, $\Delta = 0.6$ used in all measurements in the paper. To see the phase boundaries, run with `--n-simplices 20000` or larger (estimated runtime: many hours per grid point). ```bash python examples/phase_diagram.py --grid-size 10 --n-simplices 400 --n-sweeps 50 ``` ```{image} assets/cdt/phase_diagram.png :alt: Phase diagram of 4D CDT :width: 70% :align: center ``` --- ## Summary of Confirmations | Paper figure | Observable | Status | Notes | |:------------|:-----------|:------:|:------| | Fig. 2 | $N_{32}$ distribution | **Confirmed** | Sharp peaks, correct ratio $N_{32}/N_{41} \approx 2.3$ | | Figs. 9--10 | Spectral dimension $D_S(\sigma)$ | **Qualitatively confirmed** | Monotonic rise, finite-size offset | | Fig. 11 | Kinetic dimension $D_2$ | **Consistent** | Minimum near $D_2 = 2$ | | Figs. 4--6 | Volume profiles A/B/C | Not yet visible | Requires $N_4 > 10\,000$ | | Figs. 7--8 | Hausdorff dimension $D_H$ | Not yet visible | Requires $N_4 > 20\,000$ | | Fig. 3 | Phase diagram | Not yet visible | Requires $N_4 > 20\,000$ | | Figs. 12--13 | Effective action | Partially confirmed | $D_2$ minimum correct; $\cos^3$ needs larger lattice | --- ## References 1. J. Ambjorn, J. Jurkiewicz, R. Loll, *Reconstructing the Universe*, Phys. Rev. D **72** (2005), [hep-th/0505154](https://arxiv.org/abs/hep-th/0505154) 2. A. Gorlich, *Introduction to Causal Dynamical Triangulations*, Lecture notes (Zakopane, 2013) 3. R. Loll, *Quantum Gravity from Causal Dynamical Triangulations: A Review*, Class. Quant. Grav. **37** (2020), [arXiv:1905.08669](https://arxiv.org/abs/1905.08669)